Question:

For $t \in(0,2 \pi)$, if $ABC$ is an equilateral triangle with vertices$A ( \sin t, -{\cos t}), B ( \cos t, \sin t )$ and $C ( a , b )$ such that its orthocentre lies on a circle with centre $\left(1, \frac{1}{3}\right)$, then $\left(a^2-b^2\right)$ is equal to :

Updated On: Aug 4, 2024

$\frac{8}{3}$
8
$\frac{77}{9}$
$\frac{80}{9}$

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The Correct Option is B

Solution and Explanation

From the given options the correct answer is option (B): 8

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Concepts Used:

Coordinates of a Point in Space

Three-dimensional space is also named 3-space or tri-dimensional space.

It is a geometric setting that carries three values needed to set the position of an element. In Mathematics and Physics, a sequence of ‘n’ numbers can be acknowledged as a location in ‘n-dimensional space’. When n = 3 it is named a three-dimensional Euclidean space.

The Distance Formula Between the Two Points in Three Dimension is as follows;

The distance between two points P₁ and P₂ are (x₁, y₁) and (x₂, y₂) respectively in the XY-plane is expressed by the distance formula,
Distance Formula Between the Two Points in Three Dimension